rain vs come down

rain is a noun but come down is not a noun.

Word	Noun	Adjective	Verb	Adverb
rain	Yes	No	Yes	No
come down	No	No	Yes	No

As verbs, come down is a hypernym of rain; that is, come down is a word with a broader meaning than rain:

Other hypernyms of rain include fall, precipitate.

rain (noun)	come down (noun)
anything happening rapidly or in quick successive
water falling in drops from vapor condensed in the atmosphere
drops of fresh water that fall as precipitation from clouds

rain (verb)	come down (verb)
precipitate as rain	get sick
	be the essential element
	criticize or reprimand harshly
	move downward and lower, but not necessarily all the way
	fall from clouds